The homotopy theory of dg-categories and derived Morita theory

Abstract

The main purpose of this work is to study the homotopy theory of dg-categories up to quasi-equivalences. Our main result is a description of the mapping spaces between two dg-categories C and D in terms of the nerve of a certain category of (C,D)-bimodules. We also prove that the homotopy category Ho(dg-Cat) possesses internal Hom’s relative to the (derived) tensor product of dg-categories. We use these two results in order to prove a derived version of Morita theory, describing the morphisms between dg-categories of modules over two dg-categories C and D as the dg-category of (C,D)-bi-modules. Finally, we give three applications of our results. The first one expresses Hochschild cohomology as endomorphisms of the identity functor, as well as higher homotopy groups of the classifying space of dg-categories (i.e. the nerve of the category of dg-categories and quasi-equivalences between them). The second application is the existence of a good theory of localization for dg-categories, defined in terms of a natural universal property. Our last application states that the dg-category of (continuous) morphisms between the dg-categories of quasi-coherent (resp. perfect) complexes on two schemes (resp. smooth and proper schemes) is quasi-equivalent to the dg-category of quasi-coherent (resp. perfect) complexes on their product.